The Square of a Binomial

To compute (a + b)2, the square of a binomial, we can write it as
(a + b)(a + b) and use FOIL:

(a + b)2

= (a + b)(a + b)

= a2 + ab + ab + b2

= a2 + 2ab + b2

So to square a + b, we square the first term (a2), add twice the
product of the two terms (2ab), then add the square of the last term (b2).
The square of a binomial occurs so frequently that it is helpful to learn this
new rule to find it. The rule for squaring a sum is given symbolically as
follows.

The Square of a Sum

(a + b)2 = a2 + 2ab + b2

Example 1

Using the rule for squaring a sum

Find the square of each sum.

a) (x + 3)2

b) (2a + 5)2

Solution

a) (x + 3)2 =

x2

+ 2(x)(3)

+ 32

= x2 + 6x + 9

↑

↑

↑

Square of first

Twice the procuct

Square of last

b) (2a + 5)2

= (2a)2 + 2(2a)(5) + 52

= 4a2 + 20a + 25

Caution

Do not forget the middle term when squaring a sum. The equation (x + 3)2
= x2 + 6x + 9 is an identity, but (x + 3)2 = x2
+ 9 is not an identity. For example, if x = 1 in (x + 3)2 = x2
+ 9, then we get 42 = 12 + 9, which is false.

When we use FOIL to find (a - b)2, we see that

(a - b)2

= (a - b)(a - b)

= a2 - ab - ab + b2

= a2 - 2ab + b2

So to square a - b, we square the first term (a2), subtract twice
the product of the two terms (-2ab), and add the square of the last term (b2).
The rule for squaring a difference is given symbolically as follows.